"Mantra", "Yantra", and "Tantra" are terms in the Sanskrit language. Can someone with expertise define their true meaning in the context of Vedic philosophy and how the three terms may be inter-related?
As these comes under the Agama so it is not possible to discuss about it from Vedic point of view hence the agamic explanations are given below.
In general, tantra is a mode of worship which is based on the Agamas. There are also scriptures which are referred to as tantras which deal with worship. It is more of a practical approach and less of philosophy.
mantra can be a letter or word or groups of words which possess divine powers. When chanted in a proper ways it could invoke deities and hidden faculties of humans.
yantra or sigils are like geometrical figures which may or many not contain letters and numbers inside them. They are used to invoke and worship deities just like idol or a image of god.
For better understanding read this,
The primary Mantra of a Devata is called Mula-Mantra. Mantras are solar (Saura) and masculine, and lunar (Saumya) and feminine, as also neuter. If it be asked why things of mind are given sex, the answer is for the sake of the requirements of the worshipper. The masculine and neuter forms are called specifically Mantra and the feminine Vidya, though the first term may be used for both. Neuter Mantras end with Namah. Hum, Phat are masculine terminations, and "Tham" or Svaha, feminine (see Sharadatilaka II. Narada-pañcaratra VII, Prayogasara, Pranatoshini 70).
The Nitya Tantra gives various names to Mantra according to the number of the syllables such as Pinda, Kartari, Bija, Mantra, MAlA. Commonly however the term Bija is applied to monosyllabic Mantras.
The word "Mantra" comes from the root "man" to think. "Man" is the first syllable of manana or thinking. It is also the root of the word "Man" who alone of all creation is properly a Thinker. "Tra" comes from the root "tra," for the effect of a Mantra when used with that end, is to save him who utters and realizes it. Tra is the first syllable of Trana or liberation from the Samsara. By combination of man and tra, that is called Mantra which, from the religious stand-point, calls forth (Amantrana) the four aims (Caturvarga) of sentient being as happiness in the world and eternal bliss in Liberation. Mantra is thus Thought-movement vehicled by, and expressed in, speech. Its Svarupa is, like all else, consciousness (Cit) which is the Shabda-Brahman. A Mantra is not merely sound or letters. This is a form in which Shakti manifests Herself. The mere utterance of a Mantra without knowing its meaning, without realization of the consciousness which Mantra manifests is a mere movement of the lips and nothing else. We are then in the outer husk of consciousness; just as we are when we identify ourselves with any other form of gross matter which is, as it were, the "crust" (as a friend of mine has aptly called it) of those subtler forces which emerge from the Yoni or Cause of all, who is, in Herself Consciousness (Cidrupini). When the Sadhaka knows the meaning of the Mantra he makes an advance. But this is not enough. He must, through his consciousness, realize that Consciousness which appears in the form of the Mantra, and thus attain Mantra-Caitanya. At this point, thought is vitalized by contact with the center of all thinking. At this point again thought becomes truly vital and creative. Then an effect is created by the realization thus induced.
The same power which manifests to the ear in the Mantra is represented in the lines and curves of the Yantra which, the Kaulavali Tantra says, is the body of the Devata:
Yantram mantramayam proktam mantratma devataiva hi
Dehatmanor yatha bhedo yantra-devata yoshtatha.
The Yantra is thus the graphic symbol of the Shakti, indicated by the Mantra with which identification takes place. The Pratima or image is a grosser visual form of the Devata. But the Mantras are particular forms of Divine Shakti, the realization of which is efficacious to produce particular results. As in Kundalini- Yoga, so also here the identification of the Sadhaka with different Mantras gives rise to various Vibhutis or powers: for each grouping of the letters represents a new combination of the Matrika Shaktis. It is the eternal Shakti who is the life of the Mantra. Therefore, Siddhi in Mantra Sadhana is the union of the Sadhaka's